A refined cubic difference inequality with a squared Vandermonde correction

Let $x,y,z>0$ , show \begin{eqnarray*} && x(x-y)(x-z) + y(y-z)(y-x) + z(z-x)(z-y) \ &\geq& xyz \left( 1 - \frac{8xyz}{(x+y)(y+z)(z+x)} \right) + \frac{4(x-y)^2(y-z)^2(z-x)^2}{(x+y)(y+z)(z+x)} \end{eqnarray*}

Source: InternLM Lean-Workbook, record lean_workbook_plus_58991 (Apache-2.0). Complete source proposition preserved; proof developed independently.